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arXiv:2012.05265 · Physical review. B./Physical review. B

Tensor network study of the m=12 magnetization plateau in the Shastry-Sutherland model at finite temperature

Piotr Czarnik, Marek M. Rams, Philippe Corboz, Jacek Dziarmaga

Published Feb 4, 2021• 41 citations
arXivHTMLPDF

Abstract

The two-dimensional infinite projected entangled pair state tensor network is evolved in imaginary time with the full update (FU) algorithm to simulate the Shastry-Sutherland model in a magnetic field at finite temperature directly in the thermodynamic limit. We focus on the phase transition into the $m=\frac{1}{2}$ magnetization plateau, which was observed in experiments on ${\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}$. For the largest simulated bond dimension, the early evolution in the high-temperature regime is simulated with the simple update (SU) scheme and then, as the correlation length increases, continued with the FU scheme towards the critical regime. We apply a small symmetry-breaking bias field and then extrapolate towards zero bias using a simple scaling theory in the bias field. The combined SU $+$ FU scheme provides an accurate estimate of the critical temperature, even though the results could not be fully converged in the bond dimension in the vicinity of the transition. The critical temperature estimate is improved with a generalized scaling theory that combines two divergent length scales: One due to the bias, and the other due to the finite bond dimension. The obtained results are consistent with the transition being in the universality class of the two-dimensional classical Ising model. The estimated critical temperature is 3.5(2) K, which is well above the temperature 2.1 K used in the experiments.

DOI
10.1103/physrevb.103.075113

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The two-dimensional infinite projected entangled pair state tensor network is evolved in imaginary time with the full update (FU) algorithm to simulate the Shastry-Sutherland model in a magnetic field at finite temperature directly in the thermodynamic limit. We focus on the phase transition into the m=12m=\frac{1}{2}m=21​ magnetization plateau, which was observed in experiments on SrCu2(BO3)2{\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}SrCu2​(BO3​)2​. For the largest simulated bond dimension, the early evolution in the high-temperature regime is simulated with the simple update (SU) scheme and then, as the correlation length increases, continued with the FU scheme towards the critical regime. We apply a small symmetry-breaking bias field and then extrapolate towards zero bias using a simple scaling theory in the bias field. The combined SU +++ FU scheme provides an accurate estimate of the critical temperature, even though the results could not be fully converged in the bond dimension in the vicinity of the transition. The critical temperature estimate is improved with a generalized scaling theory that combines two divergent length scales: One due to the bias, and the other due to the finite bond dimension. The obtained results are consistent with the transition being in the universality class of the two-dimensional classical Ising model. The estimated critical temperature is 3.5(2) K, which is well above the temperature 2.1 K used in the experiments.
The Shastry-Sutherland model is an effective model of the layered material SrCu_{2}(BO_{3})_{2}, which exhibits an extremely rich phase diagram as a function of pressure and magnetic field. Motivated by the recent controversy regarding its phase diagram at zero magnetic field, we perform large-scale simulations based on infinite projected entangled-pair states, a two-dimensional tensor network Ansatz to represent the ground state directly in the thermodynamic limit. By employing the latest optimization techniques, we obtain variational states with lower energy than previous results obtained from other methods. Using systematic extrapolations to the exact infinite bond dimension limit, our simulations reveal a narrow quantum spin liquid phase between the plaquette and antiferromagnetic phases in the range 0.785(5)≤J^{'}/J≤0.82(1).
We investigate the thermal properties of the 13\frac{1}{3}31​ plateau in the Shastry-Sutherland model with infinite projected entangled-pair states by performing the imaginary-time evolution of the infinite-temperature density matrix. We show that both the Z2{\mathbb{Z}}_{2}Z2​ and Z3{\mathbb{Z}}_{3}Z3​ broken symmetries of the ground state are restored at a unique temperature where the correlation length has a peak, and that the melting of the plateau occurs via a single weakly first-order transition. We focus on the experimentally relevant coupling constants deep into the 13\frac{1}{3}31​ plateau phase at h=1h=1h=1 and ${J}^{\ensuremath{'}}/J=0.63$, which was estimated to describe the SrCu2(BO3)2{\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}SrCu2​(BO3​)2​ compound. By computing the free energy we are able to locate the transition temperature around ${T}_{c}\ensuremath{\simeq}4.8\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, well above the temperature T=2KT=2\phantom{\rule{0.28em}{0ex}}\mathrm{K}T=2K at which the 13\frac{1}{3}31​ plateau was observed in experiments. The investigation is supplemented by adding a bias term to the Hamiltonian and studying the induced crossover. We further map the transition line in the field-temperature phase diagram.
Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian HHH can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension DDD. The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at Tc=0.0606(4)J{\mathcal{T}}_{c}=0.0606(4)JTc​=0.0606(4)J, where JJJ is the isotropic coupling constant between S=12S=\frac{1}{2}S=21​ pseudospins.
We simulate the Shastry-Sutherland model in two dimensions by means of infinite projected entangled-pair states (iPEPS)---a variational tensor network method where the accuracy can be systematically controlled by the so-called bond dimension. Besides the well-established dimer and N\'eel phase iPEPS confirms the presence of an intermediate phase with plaquette long-range order, and we determine its phase boundaries with high accuracy. The first-order phase transition for J=0.675(2)J=0.675(2)J=0.675(2) between dimer and plaquette phase is compatible with previous series expansion results. iPEPS predicts a weak first-order phase transition between plaquette and N\'eel phase occurring for J=0.765(15)J=0.765(15)J=0.765(15). We do not find a stable intermediate columnar-dimer phase, even when we bias the state towards this order.
The Hubbard model is a longstanding problem in the theory of strongly correlated electrons and a very active one in the experiments with ultracold fermionic atoms. Motivated by current and prospective quantum simulations, we apply a two-dimensional tensor network---an infinite projected entangled pair state---evolved in imaginary time by the neighborhood tensor update algorithm working directly in the thermodynamic limit. With $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry and the bond dimensions up to 29, we generate thermal states down to the temperature of 0.17 times the hopping rate. We obtain results for spin and charge correlators, unaffected by boundary effects. The spin correlators---measurable in prospective ultracold atoms experiments attempting to approach the thermodynamic limit---provide evidence of disruption of the antiferromagnetic background with mobile holes in a slightly doped Hubbard model. The charge correlators reveal the presence of hole-doublon pairs near half filling and signatures of hole-hole repulsion on doping. We also obtain specific heat in the slightly doped regime.
We study second-order finite-temperature phase transitions of the two-dimensional quantum Ising and interacting honeycomb fermion models using infinite projected entangled pair states (iPEPS). We obtain an iPEPS thermal state representation by variational tensor network renormalization. We find that at the critical temperature Tc{T}_{c}Tc​ the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension DDD. Motivated by this observation, we investigate the application of finite correlation length scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at T=0T=0T=0, to obtain precise values of Tc{T}_{c}Tc​ and the universal critical exponents. We find that in the vicinity of Tc{T}_{c}Tc​ the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain Tc{T}_{c}Tc​ and the critical exponents in agreement with quantum Monte Carlo results except for couplings close to the quantum critical points where larger bond dimensions are required.